Permutation BASIC(Without repetition)

PERMUTATION WITHOUT REPETATION

We define the symbol ! (factorial), as follows: 0! = 1, and for integer n ≥ 1,

n! = 1 · 2 · 3 · · · n.

n! is read n factorial.

Let x₁, x₂, . . . , xₙ be n distinct objects. A permutation of these objects is simply a rear-

rangement of them.

Example 1

There are 24 permutations of the letters in MATH, 

namely

MATH, MAHT ,MTAH ,MTHA

 MHTA ,MHAT

AMTH, AMHT ,ATMH ,ATHM AHTM ,AHMT

TAMH ,TAHM ,TMAH, TMHA THMA, THAM

HATM ,HAMT, HTAM, HTMA

 HMTA ,HMAT

Theorem Let x₁, x₂, . . . , xₙ be n distinct objects. Then there are n! permutations of them.

Proof: The first position can be chosen in n ways, the second object in n − 1 ways, the third

in n − 2, etc. This gives

n(n − 1)(n − 2) · · · 2 · 1 = n!.

EXAMPLE 2 

Given a rational number,

write it as a fraction in lowest terms and calculate the

product of the resulting numerator and denominator. For

how many rational numbers between 0 and 1 will 20! be

the resulting product?[AIME 1991]

ANS 128

SOLUTION 

20! Has 8 prime factors

No of ways of arranging them are 2⁸

but the number ∈ (0,1)

So half of them are less than 1

2⁸/2=2⁷=128

Example 3

A student is to answer 10 out of 13 questions such that he must choose atleast 4 from the first 5 questions find the number of choices available to him.[AIEEE 2003 INDIA]

Ans= 196

Hint

⁵C₄ • ⁸C₆ + ⁵C₅ •⁸C₅ =196